Simplyifying factorials

Question

I am looking to simplify the factorial:

2(i!)/(2i)!.

I know that this could be simplified to:

2/(product of numbers between i+1 and 2i).

However I am unsure how to write that as a more simplified version of the original. To me, this seems as simple as possible.

Answer 1

You have

$$ \frac{2(i!)}{(2i)!}=\frac{2(1\cdot 2\cdot\ldots\cdot i)}{(1\cdot 2\cdot\ldots\cdot i)\cdot\bigl((i+1)(i+2)\dots(2i)\bigr)}. $$

I can't see more simplified form. Maybe $$2\prod_{k=1}^i \frac{1}{i+k}?$$ Of course, this is ecactly the same.

Answer 2

If it's a mathematical notation you're looking for, then you could do this:

$$\frac{2(i!)}{(2i!)}=\frac{2(1\cdot 2\cdot\ldots\cdot i)}{(1\cdot 2\cdot\ldots\cdot i)\cdot\bigl((i+1)(i+2)\dots(2i)\bigr)}=\frac{2}{\prod_{j={i+1}}^{2i} j}=2 {(\prod_{j={i+1}}^{2i} j)}^{-1}$$

There is no other more simplified notation for the simplification, I believe.